Thursday, July 26, 2012

MODULE 4 - SRINIVASA RAMANUJAN (1887 AD - 1920 AD)


SRINIVASA RAMANUJAN (1887 AD - 1920 AD)



Born
22nd December 1887 AD (1707-04-15)
 
Died
1920 AD
Residence
Erode , Kumbakonam
Nationality
Indian
Fields
Mathematics, Astronomy
Institutions
Cambridge university, madras university
Friend
Hardy

Srinivasa Ramanujan, one of India’s greatest mathematical geniuses, was born in his grandmother’s house in Erode, a small village about 400 km southwest of Madras, on 22nd December 1887. His father worked in kumbakonam as a clerk in a cloth merchant’s shop. In 1917 he was hospitalized, his doctors fearing for his life. By late 1918 his health had improved; he returned to India in 1919. But his health failed again, and he died the next year.
Ø  Five years old – primary school
Ø  Jan 1898 – town high school in Kumbakonam
Ø  1904 – he got scholarship
Ø  1906 – he entered in to Pachaiyappa’s college
Ø  14th July 1909 – he married ten year old girl S.Janaki Ammal
Ø  1911 – His first paper published, 17 page works on Bernoulli numbers - journal of the Indian Mathematical Society.
Ø  Ramanujan was appointed to the post of clerk and began his duties on 1st March 1912.
Ø  1914 – he went England
Ø  1916 – Cambridge university granted him a bachelor of science degree
Ø  1919 – he returned India






Contributions
·             Ramanujam made substantial contributions to the analytical theory of numbers and    worked on elliptic functions, continued fractions and infinite 1900 he began to work on his own on mathematics summing geometric and arithmetic series.
·             He worked on divergent series. He sent 120 theorems on imply divisibility properties of the partition function.
·             He gave a meaning to eulerian second integral for all values of n (negative, positive and fractional). He proved that the integral of xn-1 e-7 = ¡   (gamma) is true for all values of gamma.
·             Goldbach’s conjecture: Goldbach’s conjecture is one of the important illustrations of ramanujan contribution towards the proof of the conjecture. The statement is every even integer greater that two is the sum of two primes, that is, 6=3+3 : Ramanujan and his associates had shown that every large integer could be written as the sum of at most four (Example: 43=2+5+17+19).
·        Partition of whole numbers: Partition of whole numbers is another similar problem that captured ramanujan attention. Subsequently ramanujan developed a formula for the partition of any number, which can be made to yield the required result by a series of successive approximation. Example 3=3+0=1+2=1+1+1;
·             Numbers: Ramanujan studied the highly composite numbers also which are recognized as the opposite of prime numbers. He studies their structure, distribution and special forms.
·             Fermat Theorem: He also did considerable work on the unresolved Fermat theorem, which states that a prime number of the form 4m+1 is the sum of two squares.
·             Ramanujan number: 1729 is a famous ramanujan number. It is the smaller number which can be expressed as the sum of two cubes in two different ways- 1729 = 13 + 123 = 93 + 103
·             Cubic Equations and Quadratic Equation: Ramanujam was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quadratic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.
·             Euler’s constant : By 1904 Ramanujam had began to undertake deep research. He investigated the series (1/n) and calculated Euler’s constant to 15 decimal places.
·             Hypo geometric series: He worked hypo geometric series, and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions. Ramanujan’s own works on partial sums and products of hyper-geometric series have led to major development in the topic.








·             Journal of the Indian mathematical society: Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the journal of the Indian mathematical society. He developed relations between elliptic modular equations in 1910.
·             Bernoulli numbers: He published a brilliant research paper on Bernoulli numbers in 1911 in the journal of the Indian mathematical society and gained recognition for his work. Despite his lack of a university education, he was becoming well known in the madras area as a mathematical genius. He began to study the Bernoulli numbers, although this was entirely his own independent discovery.
·             Ramanujan worked out the Riemann series, the elliptic integrals hyper geometric series and functions equations of the zeta functions on the other hand he had only a vague idea of what constitutes a mathematical proof. Despite many brilliant results, some of his theorems on prime numbers were completely wrong.
·             Ramanujan independently discovered results of gauss, Kummar and others on hyper-geometric series.
·             Perhaps has most famous work was on the number p(n) for small numbers n, and ramaujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. others were only proved after Ramanujan’s death.  In a joint paper with hardly, ramanujan gave an asymptotic formulas for p(n). It had the remarkable property that it appeared to give the correct value of p(n), and this was later proved by Rademacher.
·             Ramanujan discovered a number of remarkable identities that imply divisibility properties of the partition function. He also produced quite a number of results in definite integrals in the form of general formulate.
            
           Besides his published work, ramanujan left behind several notebooks filled    with theorems that mathematicians have continued to study. The English    Mathematician G.N Watson, from 1918 to 1951, published 14 papers under the general title theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by ramanjan work. In 1997 ramanujan journal was launched to publish work in areas mathematics influenced by Ramanujan”. 




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